The aim of this paper is to present experimental results on a recently developed method of factor analysis of data with graded, or fuzzy, attributes. The method utilizes formal concepts of data with graded attributes. In our previous papers, we described the factor model, the method, an algorithm to compute factors, and provided basic examples. In this paper, we perform a more extensive experimentation with this method. In particular, we apply the method to factor analysis of sports data. The aim of the paper is to demonstrate that the method yields reasonable factors, explain in detail how the factor model and the factors are to be understood, and to put forward new issues relevant to the method.
1.1
Recently, a considerable effort was devoted to the development of factor analysis
and related methods for new types of data such as Boolean (binary) or ordinal. In
our previous papers, we developed a method of factor analysis of Boolean data
[
In a broad sense, our method may be considered as implementing the general
idea of factor analysis [
I = A ◦ B
(1)
of I into a product of an n × k object-factor matrix A, a k × m matrix B,
revealing thus k factors, i.e. new, possibly more fundamental attributes (or variables),
which explain the original m attributes. We want k < m and, in fact, k as small
as possible to achieve parsimony: The n objects described by m attributes via
I may then be described by k factors via A, with B representing a
relationship between the original attributes and the factors. Contrary to classic factor
analysis, which uses the calculus of real-valued matrices, we use the calculus of
matrices over residuated lattices. That is, the entries of matrices involved are
elements of a residuated lattice L = hL, ∧, ∨, ⊗, →, 0, 1i, i.e. Iij , Ail, Blj ∈ L. The
elements of L represent truth degrees, 0 and 1 are the smallest and largest one
and correspond to “(fully) false” and “(fully) true”; ∧ and ∨ denote the infimum
and supremum, and ⊗ and → denote the truth functions on many-valued logic
connectives of conjunction and implication. The product ◦ in (1) is defined by
(A ◦ B)ij = Wlk=1 Ail ⊗ Blj . (2)
Importantly, the entries of I, A, and B are interpreted the following way:
Iij is the truth degree of the proposition “object i has attribute j”,
Ail is the truth degree of the proposition “factor l applies to object i”,
Blj is the truth degree of “attribute j is one of the manifestations of factor l”.
For the moment, think of i, j, and l as a particular athlete (object), good
performance in long jump (attribute), and good speeding ability (factor). Using
the principles of fuzzy logic [
In principle, our method works as follows. We compute from I, using a greedy
approximation algorithm from [
F = {hC1, D1i, . . . , hCk, Dki} ⊆ B(X, Y, I) of formal fuzzy concepts of I, which gives us the decomposition as follows. Put (AF )il = (Cl)(i) and (BF )lj = (Dl)(j), i.e. AF is an n × k matrix in which the lth column consists of grades assigned to objects by the lth concept extent Cl and BF is a k × m matrix in which the lth row consists of grades assigned to attributes by the lth intent Dl. Then I = AF ◦ BF , i.e. the matrices AF and BF induced by F provide us with (4) (5)
a decomposition of I. Moreover, the optimal decompositions (i.e. with least k possible) may in principle be obtained this way, in which sense using formal concepts as factors is an optimal strategy. Note however, that our algorithm computes suboptimal decompositions since the problem to compute an optimal decomposition is an NP-hard optimization problem.
We revisit these notions, particularly in Section 2.1, where we explain the notions involved using a particular example. ✵✿✵✵ ✵✿✷ ✵✿✵ ✵✿✼ ✶✿✵✵ .
Note that due to the well-known Miller’s 7 ± 2 phenomenon [
Our aim in this section is to present the results of selected analyses, and thus to demonstrate a usefulness of the method, as well as to explain in detail the process of analysis, possibly even for a user who is not familiar with the technicalities of the method.
First, we point out some features common to the examples presented below.
As the complete residuated lattice, we use as five-element Lukasiewicz chain.
That is, the matrix degrees are taken from the set
Many other choices are available, see e.g. [
L = {0, 0.25, 0.5, 0.75, 1} a ⊗ b = max(0, a + b − 1). 2.1
2004 Olymphic Games Decathlon—Top 5
We start with a detailed description of factor analysis of top 5 athletes in the
2004 Olympic Decathlon and use this example as a reference example in the
subsequent sections (this data is also used in [
The top table in Tab. 1 contains the results of top 5 athletes according to the
IAAF scoring tables. The second table from the top contains the corresponding
matrix I, i.e. the matrix with degrees from the five-element scale L, and the
bottom table contains its graphical representation. The transformation from the
table with scores to the matrix with degrees from L = {0, 0.25, 0.5, 0.75, 1} is
accomplished using functions
p − Lj
Hj − Lj
sj : [0, . . . , 1400] → L defined by sj (p) = round
where j is an attribute (discipline), and Lj and Hj are the lowest and the highest
scores achieved by all the athletes (i.e. not only the top 5) who participated in
the competition, and round is the function rounding the numbers in [
The algorithm from [
Fig. 2 shows rectangular patterns using which the factors may be visualized. Each rectangular pattern labeled Fl is actually the matrix Jl resulting as the Cartesian product of the extent Cl and the intent Dl of Fl, i.e. we have (Jl)ij = Cl(i) ⊗ Dl(j). (For readers familiar with the ordinary FCA, let us note that these patterns are the rectangles corresponding to formal concepts and that in the general situation with degrees, the concepts cannot be uniquely restored from these patterns.)
10 lj sp hj 40 hu di pv ja 15 Sebrle 894 1020 873 915 892 968 844 910 897 680
Clay 989 1050 804 859 852 958 873 880 885 668 Karpov 975 1012 847 887 968 978 905 790 671 692
Macey 885 927 835 944 863 903 836 731 715 775
Warners 947 995 758 776 911 973 741 880 669 693 ❋✍ ❋✷ ❋✸ ❋✎ ❋✺ ❋✻ ❋✍ ❋✷ ❋✸ ❋✎ ❋✺ ❋✻ Legend: 10—100 meters sprint race; lj—long jump; sp—shot put; hj—high jump; 40—400 meters sprint race; hu—110 meters hurdles; di—discus throw; pv—pole vault; ja—javelin throw; 15—1500 meters run.
Fig. 2. Factor Concepts as Rectangular Patterns. Fi Extent Intent F1 {.5/Sebrle, Clay, Karpov, .5/Macey, .75/Warners} {10, lj, .75/sp, .75/hj, .5/40, hu, .5/di, .25/pv, .25/ja, .5/15} F2 {Sebrle, .75/Clay, .25/Karpov, .5/Macey, .25/Warners} {.5/10, lj, sp, hj, .75/40, hu, .75/di, .75/pv, ja, .75/15} F3 {.75/Sebrle, .5/Clay, .75/Karpov, Macey, .5/Warners} {.5/10, .5/lj, .75/sp, hj, .75/40, .75/hu, .75/di, .25/pv, .5/ja, 15} F4 {Sebrle, .75/Clay, .75/Karpov, .75/Macey, Warners} {.5/10, .75/lj, .5/sp, .5/hj, .75/40, hu, .25/di, .5/pv, .25/ja, .75/15} F5 {.75/Sebrle, .5/Clay, Karpov, .75/Macey, .25/Warners} {.75/10, .75/lj, sp, .75/hj, 40, hu, di, .25/pv, .25/ja, .75/15} F6 {.75/Sebrle, Clay, .25/Karpov, .5/Macey, .25/Warners} {.75/10, lj, .75/sp, .75/hj, .5/40, hu, di, .5/pv, ja, .5/15} Fig. 3 demonstrates what portion of matrix I is explained using the first l factors for l = 1, . . . , k. In particular, the matrix labeled 56% just shows the rectangular pattern J1 corresponding to F1. The number indicates that 56% of the entries in I have the same value as in J1, i.e. 56% of the data is explained by the first factor. The second matrix contains J1 ∨ J2, i.e. it illustrates what happens when we add the second factor. As we can see, 82% of the data is explained by the first two factors. Since the first three factors explain 91% of the data, one might say that the first three factors account for most of the data, are most important, and the rest of the factors may be omitted. Nevertheless, adding further the factors we see that the first four, five, and six factors explain 95%, 98%, and 100% of the data (the latter fact is obviously true because the factors completely decompose matrix I). Note also, that several of the 18% = 100% − 82% of the entries not explained by the first two factors have values close to the corresponding entries of I, so a measure of closeness of Jl and I which takes into account also close entries, rather than exactly equal ones only, would yield a number larger than 82%. In any case, we can conclude from the visual inspection of the matrices that already the first two or three factors explain the data reasonably well. Note that the fact that the revealed factors are reasonable was confirmed to us by an experienced decathlon coach who also pointed out to us that F2 (explosiveness) is known to be well-developed by the Czech school of decathlon (hence Sebrle).
56% 95% 82% 98% 91% 100%
Let us turn to the interpretation of the factors. For this purpose, Fig. 2 is crucial since it contains all the information about the factors. Note however that Fig. 2 is also helpful as it shows the clusters corresponding to the factor concepts which draw together the athletes and their performances in the events. Factor F1: F1 applies to Sebrle to degree 0.5, to both Clay and Karpov to degree 1, to Macey to degree 0.5, and to Warners to degree 0.75. Furthermore, this factor applies to attribute 10 (100 m) to degree 1, to attribute lj (long jump) to degree 1, to attribute sp (shot put) to degree 0.75, etc. This means that an excellent performance (degree 1) in 100 m, an excellent performance in long jump, a very good performance (degree 0.75) in shot put, etc. are particular manifestations of this factor. On the other hand, only a relatively weak performance (degree 0.25) in javelin throw and pole vault are manifestations of this factor. All the manifestations of this factor with degree 1 are 100 m, long jump, and 110 m hurdles. This factor can be interpreted as the ability to run fast for short distances. Note that this factor applies particularly to Clay and Karpov which is well known in the world of decathlon. Factor F2: Similarly, since the manifestations of this factor with degree 1 are long jump, shot put, high jump, and javelin, F2 can be interpreted as the ability to apply very high force in a very short term (explosiveness). F2 applies particularly to Sebrle, and then to Clay, who are known for this ability. Factor F3: Manifestations with grade 1 are high jump and 1500 m. This factor is typical for lighter, not very muscular athletes. Macey, who is evidently that type among decathletes (196 cm and 98 kg) is the athlete to whom the factor applies to degree 1. These are the most important factors behind data matrix I. 2.2
2004 Olympic Games Decathlon Top 5 By Their Best Results In this example, we take the top 5 athletes of the 2004 Olympic Decathlon but we take their best performances during their decathlon competitions, instead of their actual performances in a single event such as the 2004 Olympics. Taking best performances may be reasonable if we want to avoid a possible bad luck in a particular discipline such as a bad start in 100 m. Tab. 3 contains the scores. The corresponding matrix I and its decomposition into AF ◦ BF is depicted in Fig. 4. Here, the transformation from points to degrees is defined as follows. For discipline j, we put 1 for p ∈ [Hj , Hj − 100), 0.75 for p ∈ [Hj − 100, Hj − 200), sj (p) = 0.5 for p ∈ [Hj − 200, Hj − 300), 0.25 for p ∈ [Hj − 300, Hj − 400), 0 for p ≤ Hj − 400, where Hj is the highest score ever achieved during a decathlon competition for discipline j. Note that H10 = 1042; Hlj = 1117; Hsp = 1048; Hhj = 1061; H40 = 1025; Hhu = 1064; Hdi = 993; Hpv = 1152; Hja = 1040; H15 = 963.
It seems natural that the factors in this case are different from those in the example in Section 2.1. Nevertheless, we can see that F1 applies to degree 1 to Clay and Karpov in both examples and applies to the other athletes to similar degrees in both examples as well. Nevertheless, the intents of the first factor are different although a reasonable similarity is apparent as well (presence of long
10 lj sp hj 40 hu di pv ja 15 Sebrle 942 1089 880 944 921 1002 859 972 907 798
Clay 1010 1050 868 887 944 1022 993 941 920 670 Karpov 931 1073 910 915 968 984 929 1004 743 729
Macey 940 1002 841 944 998 931 836 849 799 990 Warners 947 1022 800 831 978 973 824 886 692 693 ❋✍ ❋✷ ❋✸ ❋✎ ❋✺ ❋✍ ❋✷ ❋✸ ❋✎ ❋✺ jump and hurdles to degree 1, presence of 100 m and high jump to high degrees).
A similar observation can be made on F2 (connects Sebrle and Clay) and F3 which is typical of Macey. The results of the 5th–10th athletes in the 2004 Olympic Decathlon are depicted in Tab. 4. The matrix I corresponding to the top 10 athletes, along with a decomposition I = AF ◦ BF computed by the algorithm is depicted in Fig. 5.
The same transformation from scores to degrees was used as in Section 2.1.
Compared to the factors from Section 2.1, the factors in this example are generally different although some similarities are apparent. For example, factor ✶ ✹ ❧✁ ✂s ❤✁ ❤✉ ❞✄ ✂✈ ✁✄ ☎✶ Factor analysis of sports data via decomposition of matrices with grades 301 ❋✓ ❋✷ ❋✸ ❋✔ ❋✺ ❋✻ ❋✼ ❋✽ ❋✾ ❋✓✕ ❋✓✓ ❋✓✷ ❋✓ ❋✷ ❋✸ ❋✔ ❋✺ ❋✻ ❋✼ ❋✽ ❋✾ ❋✓✕ ❋✓✓ ❋✓✷
F2 here is exactly the same (has same intent) as F1 in Section 2.1, F12 is the same as F6 in Section 2.1, and F4 is almost the same as F3 in Section 2.1.
We might nevertheless be interested in the question of how well the factors from Section 2.1 explain the new dataset regarding the top 10 athletes. A reasonable way to proceed is the following. Consider the set of 6 concepts of the new, 10 × 10 matrix I, namely,
G = {G1 = hP1, Q1i, . . . , G6 = hP6, Q6i} obtained from the factors F1 = hC1, D1i, . . . , F6 = hC6, D6i by
P1 = D1↓, Q1 = P1↑, . . . , P6 = D6↓, Q6 = P6↑, i.e. every factor Gl is the concept of the 10 × 10 matrix I generated by the intent of Fl. This way, we do not have I = AG ◦ BG in general, as can be seen from this example. Nevertheless, the first factor G1 explains 50% of the data, the first two factors 69%, the first three factors 80%, the first four factors 86%, the first five factors 89%, and all factors in G explain 91% of the data. Hence, one may conclude that the factors of the top 5 athletes explain reasonably well also the results of all the top 10 athletes. The matrices involved are depicted in Fig. 6.
Note that one may clearly observe the similarity between I (the original matrix) and AG ◦ BG (the matrix reconstructed from the factors in G). 2.4
2004 Olympic Games Modern Pentathlon Another sport that contains several disciplines and may be interesting for factor analysis is the modern pentathlon. The five disciplines are, however, rather diverse and it is therefore challenging to think of natural factors in this sport. Recall that modern pentathlon consists of pistol shooting, fencing, 200 m freestyle swimming, show jumping, and a 3 km cross-country run. Except for the fencing competition, athletes do not directly compete against one another in the five Radim Belohlavek and Marketa Krmelova ❋✓ ❋✷ ❋✸ ❋✔ ❋✺ ❋✻ ❋✓ ❋✷ ❋✸ ❋✔ ❋✺ ❋✻ events. Instead, a better absolute performance results in a higher score and the sum of all the scores for the disciplines gives the overall total score of a given athlete.
Tab. 5 contains the results of the 2004 Olympic Games modern pentathlon of the top 10 athletes. The corresponding matrix I and its decomposition into AF ◦ BF is depicted in Fig. 7. To transform the scores of discipline j to degrees, we used the function 1 for p ∈ [Hj , Hj − 15 (Hj − Lj )), 0.75 for p ∈ [Hj − 51 (Hj − Lj ), Hj − 52 (Hj − Lj )), sj (p) = 0.5 for p ∈ [Hj − 25 (Hj − Lj ), Hj − 53 (Hj − Lj )), 0.25 for p ∈ [Hj − 35 (Hj − Lj ), Hj − 54 (Hj − Lj )), 0 for p ≤ Hj − 45 (Hj − Lj ), where Hj and Lj are the highest and the lowest score achieve in discipline j in the 2004 Olympic Games modern pentathlon. Note that Hsh = 1168, Lsh = 892; Hfe = 1000, Lfe = 664; Hsw = 1376, Lsw = 1140; Hri = 1172, Lri = 584; Hru = 1116, Lru = 752.
✿❀ ✿❁ ✿❂ ✿❃ ✿❄ ✿❅ ✿❀ ✿❁ ✿❂ ✿❃ ✿❄ ✿❅
✕✖ ❢✗ ✘✕ r✙ r✚
Fig. 7. Decomposition I = AF ◦ BF .
Note that of all the factors computed, F2 is probably most interesting because it is actually known in the world of modern pentathlon. Namely, F2’s manifestations are riding and cross-country run which is typical for athletes who are in a good physical shape and have good endurance. Each of the other factors more or less corresponds to a single discipline which corresponds to the intuitive idea that the disciplines are diverse and require diverse skills. 3
Concusions, Further Issues and Future Work We presented several examples of factor analysis of sports data using a recently developed method that utilizes formal concepts as factors. Our main aim was to explain the method, to illustrate the key notions used in the method, and to demonstrate how one can understand the results of the method. It turns out from the examples that the method yields reasonable factors and that the results of the method are easy to understand.
Due to the limited scope of this paper, we presented only a limited number of examples and limited comments on the presented examples. In addition to the examples presented in this paper, we performed factor analyses of further decathlon data (namely, the World Championships), figure skating, and ice hockey players performance. We refrained from formalizing some of the issues involved, such as “explanation of data by factors”, “similarity of factors”, how well the factors of one dataset serve as good factors of another dataset, etc., and used these notions with their informal meaning only. We therefore also skipped theoretical results regarding these notions, as well as further notions and results that may help us answer further natural questions that arise in the context of the presented method, such as the influence of the choice of the scale of degrees, the operation ⊗, the influence of the transformation from scores to degrees, and the like.
More examples with detailed comments as well as a detailed treatment of
some of issues mentioned above will appear in the full version of this paper. An
interesting question that is to be a subject of our future research is a
comparison, experimental and possibly also theoretical, of relationships of the presented
method with related methods that involve matrix decomposition, notable the
non-negative matrix factorization [